Some Bazilevič functions of order beta (Q809218)

For \(\alpha >0\) and \(0\leq \beta <1\) let B(\(\alpha\),\(\beta\)) denote the set of functions f that are analytic in the open unit disk and satisfy \(f(0)=0\), \(f'(0)=1\) and \(Re\{z^{1-\alpha}f'(z)/f(z)^{1- \alpha}\}>\beta\) for \(| z| <1\). For each \(\alpha\) and \(\beta\) this set is a subclass of the family of Bazilevich functions and hence consists of univalent functions. The author determines the sharp upper and lower bounds on \(| f(z)|\) and on \(| f'(z)|\) where \(| z| =r\), \(0<r<1\), and f varies over B(\(\alpha\),\(\beta\)). Also the sharp upper bounds are obtained for the Taylor series coefficients of functions in B(\(\alpha\),\(\beta\)) for each \(\alpha\) of the form \(\alpha =1/n\) where n is a positive integer.